Homology of RP(2)

I'm currently learning some homology theory but I have some difficulties computing homology groups of a few simple spaces. If someone could do the explicit calculation for RP(2), it would be really nice.

the universal covering space is the 2 sphere, so pi 1 is Z/2Z, hence also the first homology group. that does it since the space is connected non orientable manifold so the zeroth homology is Z and the second homology i guess is zero.

i am just recalling this from 40 years ago since they don't let me teach topology for some reason, so i could be wrong.